Spearman 等级相关性的检验统计量

Question

my problem is that I have calculated Spearman rank correlations between two variables. One reviewer asked me if I could add also test statistics for all coefficients where p < 0.001.

Here is one result:

> cor.test(pl$data, pl$rang, method= "spearman")


#       Spearman's rank correlation rho

data:  pl$data and pl$rang
S = 911164.6, p-value = 1.513e-05
alternative hypothesis: true rho is not equal to 0 
sample estimates:
rho 
-0.3347658 

Is the test statistics equal to S = 911164.6 ? Is it OK that it is so big number? Sorry in advance if the question is not very professional but I spend quite some time searching for the answers in the books and on internet. :( Thank you for the help in advance.

Answer 1

Yes. The ?cor.test help page (in the Value section) describes the return value from cor.test as:

 A list with class '"htest"' containing the following components:

statistic: the value of the test statistic.

Adapting the example on that page, we see

x <- c(44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1)
y <- c( 2.6,  3.1,  2.5,  5.0,  3.6,  4.0,  5.2,  2.8,  3.8)
(result <- cor.test(x, y, method = "spearman"))
#         Spearman's rank correlation rho

# data:  x and y 
# S = 48, p-value = 0.0968
# alternative hypothesis: true rho is not equal to 0 
# sample estimates:
# rho 
# 0.6 
result$statistic
#  S 
# 48 

The statistic is given by (n ^ 3 - n) * (1 - r) / 6 where n is the length of x and r <- cor(rank(x), rank(y)) .

Answer 2

The best answer I have found is on the page: Rpubs Spearman Rank Correlation

This question is also on Cross Validated "Interpreting the Spearman's Rank Correlation Coefficient output in R. What is 'S'?"

While the forumla (n ^ 3 - n) * (1 - r) / 6 , were n is equal to the sample size of variables and r is equal to the correlation coefficient (rho) for calculting the S statistic is clear, there is no clear explaination on how to interpret the results. I have yet to find a clear answer :(